374 XV NORMED ALGEBRAS AND SPECTRAL THEORY For all x, y in A, the mapping V\-*(V* ng(x)\ng(y)) is a continuous linear form on j/'g and so, by virtue of the isomorphism of ^C(S^) and j#'g, there exists (13.1) a (complex) measure \jLXy}> on S^ such that (15.9.2.4) g(zx, y) - (Ug(z) * ng(z) \ ng(y)) = f /(z) rf/1,,,00 Js'0 for all 2- e A. When OeS0, we have seen (15.9.2.2) that the function z : /H->/(Z) is zero at the point 0; consequently, if mXty is the measure induced on Sg by HXty (13.1.8), we can write in all cases (15.9.2.5) g(zx, y) It is clear that the measures mXt y are bounded. Moreover, for given jc and y in A, the measure mXty is the only bounded measure ra' on Sg such that g(zx> y) = J z(%) dm'(%) for all z 6 A, because the functions z are dense in*g(S,)(1/.20.6). If the formula (15.9.2.1) is true, then for all x, y, z in A we must have g(zx, y) = f 2(x)x(x)$(x) dmg(i). Comparing this with (15.9.2.5), what we JSy shall in fact prove is the existence of a measure mg on S^ such that (15.9.2.6) mXiy=x$'mg for all jc, y in A (cf. (13.1.5)). If we have constructed such a measure, then for all functions F e we shall have (15.9.2.7) F • mx>y = (fltf ) • mg = (tf) - mF , where we have put mF = F • IH,, which is a bounded measure (13.14.4). We shall begin by constructing a linear mapping Fh-»wF of ^Tc(Sff) into M£(S0), satisfying the equality (15.9.2.8) F-m^.^-mp for all x, y in A, and such that mF ^ 0 whenever F S| 0. Having achieved this, the measure mg will be simply the linear form FH*WIF(!) (13.3.1).n each case, Sg is separable,